Memristors as emergent nano-electronic devices have been successfully fabricated and used in non-conventional and neuromorphic computing systems in the last years. Several behavioral or physical based models have been developed to explain their operation and to optimize their fabrication parameters. All existing memristor models are trade-offs between accuracy, universality and realism, but, to the best of our knowledge, none of them is purely characterized as quantum mechanical, despite the fact that quantum mechanical processes are a major part of the memristor operation. In this paper, we employ quantum mechanical methods to develop a complete and accurate filamentary model for the resistance variation during memristor's operating cycle. More specifically, we apply quantum walks to model and compute the motion of atoms forming the filament, tight-binding Hamiltonians to capture the filament structure and the Non-Equilibrium Green's Function (NEGF) method to compute the conductance of the device. Furthermore, we proceeded with the parallelization of the overall model through Graphical Processing Units (GPUs) to accelerate our computations and enhance the model's performance adequately. Our simulation results successfully reproduce the resistive switching characteristics of memristors devices, match with existing fabricated devices experimental data, prove the efficacy and robustness of the proposed model in terms of multi-parameterization, and provide a new and useful insight into its operation.